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cyclotomic fields


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1: 24.17 Mathematical Applications
§24.17(iii) Number Theory
Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and L -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); p -adic analysis (Koblitz (1984, Chapter 2)). …
2: Bibliography W
  • L. C. Washington (1997) Introduction to Cyclotomic Fields. 2nd edition, Springer-Verlag, New York.
  • E. Witten (1987) Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.
  • 3: 25.17 Physical Applications
    See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). … Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect). …
    4: 16.22 Asymptotic Expansions
    For asymptotic expansions of Meijer G -functions with large parameters see Fields (1973, 1983).
    5: Charles W. Clark
    He is coeditor of the following books: Atomic Excitation and Recombination in External Fields (with M. … Nayfeh), published by Gordon and Breach in 1985, Atomic Spectra and Collisions in External Fields (with M. … Taylor), published by Plenum Press in 1988, and Atoms in Strong Fields (with M. …
    6: Qiming Wang
    She has applied VRML and X3D techniques to several different fields including interactive mathematical function visualization, 3D human body modeling, and manufacturing-related modeling. …
    7: Bibliography B
  • E. Barouch, B. M. McCoy, and T. T. Wu (1973) Zero-field susceptibility of the two-dimensional Ising model near T c . Phys. Rev. Lett. 31, pp. 1409–1411.
  • R. Becker and F. Sauter (1964) Electromagnetic Fields and Interactions. Vol. I, Blaisdell, New York.
  • J. Buhler, R. Crandall, R. Ernvall, T. Metsänkylä, and M. A. Shokrollahi (2001) Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comput. 31 (1-2), pp. 89–96.
  • P. L. Butzer, S. Flocke, and M. Hauss (1994) Euler functions E α ( z ) with complex α and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.
  • 8: 14.31 Other Applications
    §14.31(i) Toroidal Functions
    Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). …
    9: 27.19 Methods of Computation: Factorization
    These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …
    10: 15.18 Physical Applications
    More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).